Civil Engineering Reference
In-Depth Information
the structure. For structures possessing a high degree of statical indeterminacy the
solution, by hand, of a large number of simultaneous equations is not practicable. The
equations would then be expressed in matrix form and solved using a computer-based
approach. Thus for a structure having a degree of statical indeterminacy equal to
n
there would be
n
compatibility equations of the form
v
1,0
+
a
11
R
1
+
a
12
R
2
+···+
a
1
n
R
n
=
0
.
v
n
,0
+
a
n
1
R
1
+
a
n
2
R
2
+···+
a
nn
R
n
=
0
or, in matrix form
.
/
!
.
/
!
v
1,0
.
v
n
,0
a
11
a
12
...
a
1
n
R
1
.
R
n
.
=−
0
"
0
"
a
n
1
a
n
2
...
a
nn
Note that here
n
is
n
s
, the degree of statical indeterminacy; the subscript 's' has been
omitted for convenience.
Alternative methods of solution of continuous beams are the slope-deflection method
described in Section 16.9 and the iterative moment distribution method described in
Section 16.10. The latter method is capable of producing relatively rapid solutions for
beams having several spans.
16.5 S
TATICALLY
I
NDETERMINATE
T
RUSSES
A truss may be internally and/or externally statically indeterminate. For a truss that is
externally statically indeterminate, the support reactions may be found by the methods
described in Section 16.4. For a truss that is internally statically indeterminate the
flexibility method may be employed as illustrated in the following examples.
E
XAMPLE
16.8
Determine the forces in the members of the truss shown in
Fig. 16.18(a); the cross-sectional area,
A
, and Young's modulus,
E
, are the same
for all members.
The truss in Fig. 16.18(a) is clearly externally statically determinate but, from Eq.
(16.5), has a degree of internal statical indeterminacy equal to 1 (
M
4). We
therefore release the truss so that it becomes statically determinate by 'cutting' one
of the members, say BD, as shown in Fig. 16.18(b). Due to the actual loads (
P
in this
case) the cut ends of the member BD will separate or come together, depending on
whether the force in the member (before it was cut) was tensile or compressive; we
shall assume that it was tensile.
=
6,
N
=
